Inflection Points

So our task is to find where a curve goes from concave upward to concave downward (or vice versa).

Calculus

The derivative of a function gives the slope.

The second derivative tells us if the slope increases or decreases.

• When the second derivative is positive, the function is concave upward.
• When the second derivative is negative, the function is concave downward.

And the inflection point is where it goes from concave upward to concave downward (or vice versa).

Example: y = 5x 3 + 2x 2 − 3x

Let's work out the second derivative:

• The derivative is y' = 15x 2 + 4x − 3
• The second derivative is y'' = 30x + 4

And 30x + 4 is negative up to x = −4/30 = −2/15, positive from there onwards. So:

And the inflection point is at x = −2/15

A Quick Refresher on Derivatives

In the previous example we took this:

y = 5x 3 + 2x 2 − 3x

and came up with this derivative:

y' = 15x 2 + 4x − 3

There are rules you can follow to find derivatives. We used the "Power Rule":

• x 3 has a slope of 3x 2 , so 5x 3 has a slope of 5(3x 2 ) = 15x 2
• x 2 has a slope of 2x, so 2x 2 has a slope of 2(2x) = 4x
• The slope of the line 3x is 3

Another example for you:

Example: y = x 3 − 6x 2 + 12x − 5

The derivative is: y' = 3x 2 − 12x + 12

The second derivative is: y'' = 6x − 12

And 6x − 12 is negative up to x = 2, positive from there onwards. So:

And the inflection point is at x = 2: